Chromatic Genus - Tunings of The Chromatic

Tunings of The Chromatic

Theon of Smyrna gives an incomplete account of Thrasyllus of Mendes' formulation of the Greater Perfect System, from which the diatonic and enharmonic genera can be deduced. For the chromatic genus, however, all that is given is a 32:27 proportion of mese to lichanos. This leaves 9:8 for the pyknon, but there is no information at all about the position of the chromatic parhypate and therefore of the division of the pyknon into two semitones, though it may have been the limma of 256:243, as Boethius does later (Barbera 1977, 306, 309). Someone has referred to this speculative reconstructions as the traditional Pythagorean tuning of the chromatic genus:

hypate parhypate lichanos mese 4/3 81/64 32/27 1/1 | 256/243 | 2187/2048 | 32/27 | -498 -408 -294 0 cents Play

Archytas used the simpler and more consonant 9/7, which he used in all three of his genera. His chromatic division is (Barbera 2001):

hypate parhypate lichanos mese 4/3 9/7 32/27 1/1 | 28/27 | 243/224 | 32/27 | -498 -435 -294 0 cents

According to Ptolemy's calculations, Didymus's chromatic has only 5-limit intervals, with the smallest possible numerators and denominators (Richter 2001). Also note that the successive intervals are all superparticular numbers:

hypate parhypate lichanos mese 4/3 5/4 6/5 1/1 | 16/15 | 25/24 | 6/5 | -498 -386 -316 0 cents